How to define the size of block?

[ysh329]

Below is a piece of CPU GEMM(4x4_11) code.

/* Create macros so that the matrices are stored in column-major order */

#define A(i,j) a[ (j)*lda + (i) ]
#define B(i,j) b[ (j)*ldb + (i) ]
#define C(i,j) c[ (j)*ldc + (i) ]

/* Block sizes */
#define mc 256
#define kc 128

#define min( i, j ) ( (i)<(j) ? (i): (j) )

/* Routine for computing C = A * B + C */

void AddDot4x4( int, double *, int, double *, int, double *, int );

void MY_MMult( int m, int n, int k, double *a, int lda, 
                                    double *b, int ldb,
                                    double *c, int ldc )
{
  int i, j, p, pb, ib;

  /* This time, we compute a mc x n block of C by a call to the InnerKernel */

  for ( p=0; p<k; p+=kc ){
    pb = min( k-p, kc );
    for ( i=0; i<m; i+=mc ){
      ib = min( m-i, mc );
      InnerKernel( ib, n, pb, &A( i,p ), lda, &B(p, 0 ), ldb, &C( i,0 ), ldc );
    }
  }
}

void InnerKernel( int m, int n, int k, double *a, int lda, 
                                       double *b, int ldb,
                                       double *c, int ldc )
{
  int i, j;

  for ( j=0; j<n; j+=4 ){        /* Loop over the columns of C, unrolled by 4 */
    for ( i=0; i<m; i+=4 ){        /* Loop over the rows of C */
      /* Update C( i,j ), C( i,j+1 ), C( i,j+2 ), and C( i,j+3 ) in
	 one routine (four inner products) */

      AddDot4x4( k, &A( i,0 ), lda, &B( 0,j ), ldb, &C( i,j ), ldc );
    }
  }
}

#include <mmintrin.h>
#include <xmmintrin.h>  // SSE
#include <pmmintrin.h>  // SSE2
#include <emmintrin.h>  // SSE3

typedef union
{
  __m128d v;
  double d[2];
} v2df_t;

void AddDot4x4( int k, double *a, int lda,  double *b, int ldb, double *c, int ldc )
{
  /* So, this routine computes a 4x4 block of matrix A

           C( 0, 0 ), C( 0, 1 ), C( 0, 2 ), C( 0, 3 ).  
           C( 1, 0 ), C( 1, 1 ), C( 1, 2 ), C( 1, 3 ).  
           C( 2, 0 ), C( 2, 1 ), C( 2, 2 ), C( 2, 3 ).  
           C( 3, 0 ), C( 3, 1 ), C( 3, 2 ), C( 3, 3 ).  

     Notice that this routine is called with c = C( i, j ) in the
     previous routine, so these are actually the elements 

           C( i  , j ), C( i  , j+1 ), C( i  , j+2 ), C( i  , j+3 ) 
           C( i+1, j ), C( i+1, j+1 ), C( i+1, j+2 ), C( i+1, j+3 ) 
           C( i+2, j ), C( i+2, j+1 ), C( i+2, j+2 ), C( i+2, j+3 ) 
           C( i+3, j ), C( i+3, j+1 ), C( i+3, j+2 ), C( i+3, j+3 ) 
	  
     in the original matrix C 

     And now we use vector registers and instructions */

  int p;
  v2df_t
    c_00_c_10_vreg,    c_01_c_11_vreg,    c_02_c_12_vreg,    c_03_c_13_vreg,
    c_20_c_30_vreg,    c_21_c_31_vreg,    c_22_c_32_vreg,    c_23_c_33_vreg,
    a_0p_a_1p_vreg,
    a_2p_a_3p_vreg,
    b_p0_vreg, b_p1_vreg, b_p2_vreg, b_p3_vreg; 

  double 
    /* Point to the current elements in the four columns of B */
    *b_p0_pntr, *b_p1_pntr, *b_p2_pntr, *b_p3_pntr; 
    
  b_p0_pntr = &B( 0, 0 );
  b_p1_pntr = &B( 0, 1 );
  b_p2_pntr = &B( 0, 2 );
  b_p3_pntr = &B( 0, 3 );

  c_00_c_10_vreg.v = _mm_setzero_pd();   
  c_01_c_11_vreg.v = _mm_setzero_pd();
  c_02_c_12_vreg.v = _mm_setzero_pd(); 
  c_03_c_13_vreg.v = _mm_setzero_pd(); 
  c_20_c_30_vreg.v = _mm_setzero_pd();   
  c_21_c_31_vreg.v = _mm_setzero_pd();  
  c_22_c_32_vreg.v = _mm_setzero_pd();   
  c_23_c_33_vreg.v = _mm_setzero_pd(); 

  for ( p=0; p<k; p++ ){
    a_0p_a_1p_vreg.v = _mm_load_pd( (double *) &A( 0, p ) );
    a_2p_a_3p_vreg.v = _mm_load_pd( (double *) &A( 2, p ) );

    b_p0_vreg.v = _mm_loaddup_pd( (double *) b_p0_pntr++ );   /* load and duplicate */
    b_p1_vreg.v = _mm_loaddup_pd( (double *) b_p1_pntr++ );   /* load and duplicate */
    b_p2_vreg.v = _mm_loaddup_pd( (double *) b_p2_pntr++ );   /* load and duplicate */
    b_p3_vreg.v = _mm_loaddup_pd( (double *) b_p3_pntr++ );   /* load and duplicate */

    /* First row and second rows */
    c_00_c_10_vreg.v += a_0p_a_1p_vreg.v * b_p0_vreg.v;
    c_01_c_11_vreg.v += a_0p_a_1p_vreg.v * b_p1_vreg.v;
    c_02_c_12_vreg.v += a_0p_a_1p_vreg.v * b_p2_vreg.v;
    c_03_c_13_vreg.v += a_0p_a_1p_vreg.v * b_p3_vreg.v;

    /* Third and fourth rows */
    c_20_c_30_vreg.v += a_2p_a_3p_vreg.v * b_p0_vreg.v;
    c_21_c_31_vreg.v += a_2p_a_3p_vreg.v * b_p1_vreg.v;
    c_22_c_32_vreg.v += a_2p_a_3p_vreg.v * b_p2_vreg.v;
    c_23_c_33_vreg.v += a_2p_a_3p_vreg.v * b_p3_vreg.v;
  }

  C( 0, 0 ) += c_00_c_10_vreg.d[0];  C( 0, 1 ) += c_01_c_11_vreg.d[0];  
  C( 0, 2 ) += c_02_c_12_vreg.d[0];  C( 0, 3 ) += c_03_c_13_vreg.d[0]; 

  C( 1, 0 ) += c_00_c_10_vreg.d[1];  C( 1, 1 ) += c_01_c_11_vreg.d[1];  
  C( 1, 2 ) += c_02_c_12_vreg.d[1];  C( 1, 3 ) += c_03_c_13_vreg.d[1]; 

  C( 2, 0 ) += c_20_c_30_vreg.d[0];  C( 2, 1 ) += c_21_c_31_vreg.d[0];  
  C( 2, 2 ) += c_22_c_32_vreg.d[0];  C( 2, 3 ) += c_23_c_33_vreg.d[0]; 

  C( 3, 0 ) += c_20_c_30_vreg.d[1];  C( 3, 1 ) += c_21_c_31_vreg.d[1];  
  C( 3, 2 ) += c_22_c_32_vreg.d[1];  C( 3, 3 ) += c_23_c_33_vreg.d[1]; 
}

[ysh329]

According to some simple analysis, you can find the block partition need to greatly utilize the cache resource. Besides, the concrete block size can be searched directly. Due to regardless of for-loop length in inner kernel, the lengths of outer for-loops are casual.